A282537 - OEIS

%I #22 Jun 12 2019 10:33:36

%S 1,0,1,0,1,1,1,1,2,1,2,2,3,2,4,3,4,4,5,5,7,6,8,8,9,9,12,11,14,14,16,

%T 16,20,19,23,24,27,27,32,32,37,38,43,44,51,51,58,61,67,69,78,80,89,93,

%U 102,106,118,121,134,140,153,159,175,181,198,207,224,234

%N Coefficients of the '5th-order' mock theta function Psi(q) with a(0)=1.

%C In Ramanujan's lost notebook the generating function is denoted by psi(q) on pages 18 and 20, however on page 20 there is a "-1" first term.

%D Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, pp. 18, 20

%H G. C. Greubel, <a href="/A282537/b282537.txt">Table of n, a(n) for n = 0..1000</a>

%F G.f.: Sum_{k>=0} x^(5*k^2) / ((1 - x^2) * (1 - x^3) * (1 - x^7) * (1 - x^8)...(1 - x^(5*k+2))).

%F G.f.: ( Sum_{k in Z} -(-1)^k * x^(5*k*(3*k + 1)/2 - 2) / (1 - x^(5*k - 2)) ) / ( Sum_{k in Z} (-1)^k * x^(5*k*(3*k - 1)) ).

%F a(n) ~ exp(Pi*sqrt(2*n/15)) / (5^(3/4)*sqrt(2*phi*n)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - _Vaclav Kotesovec_, Jun 12 2019

%e G.f. = 1 + x^2 + x^4 + x^5 + x^6 + x^7 + 2*x^8 + x^9 + 2*x^10 + 2*x^11 + ...

%t a[ n_] := If[ n < 0, 0, SeriesCoefficient[ Sum[ x^(5 k^2) / (QPochhammer[ x^2, x^5, k + 1] QPochhammer[ x^3, x^5, k]) // FunctionExpand, {k, 0, Sqrt[n/5]}], {x, 0, n}]];

%t a[ n_] := If[ n < 0, 0, With[ {m = Sqrt[1 + 24(n + 2)/5]}, SeriesCoefficient[ Sum[ -(-1)^k x^(5 k (3 k + 1)/2 - 2) / (1 - x^(5 k - 2)), {k, Quotient[m + 1, -6], Quotient[m - 1, 6]}] / QPochhammer[ x^5], {x, 0, n}]]];

%o (PARI) {a(n) = if( n<0, 0, polcoeff( sum(k=0, sqrtint(n\5), x^(5*k^2) / prod(i=1, 5*k+2, 1 - if( i%5==2 || i%5==3, x^i), 1 + x * O(x^(n - 5*k^2)))), n))};

%o (PARI) {a(n) = my(A, m); if( n<0, 0, m = sqrtint(1 + 24*(n+2)\5); A = x * O(x^n); polcoeff( sum(k=(m + 1)\-6, (m - 1)\6, -(-1)^k * x^(5*k*(3*k + 1)/2 - 2) / (1 - x^(5*k - 2)), A) / eta(x^5 + A), n))};

%Y Essentially the same as A053267.

%K nonn

%O 0,9

%A _Michael Somos_, Feb 18 2017